A377044 - OEIS

A377044

The n-th perfect-power A001597(n) minus the n-th prime-power A246655(n).

-1, 1, 4, 4, 9, 17, 18, 21, 23, 33, 47, 62, 77, 96, 98, 99, 113, 137, 159, 175, 182, 196, 207, 236, 265, 282, 297, 333, 370, 411, 433, 448, 493, 536, 579, 628, 681, 734, 791, 848, 879, 899, 962, 1028, 1094, 1159, 1192, 1220, 1293, 1364, 1437, 1514, 1559, 1591

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OFFSET

1,3

COMMENTS

Perfect-powers (A001597) are numbers with a proper integer root.

LINKS

Table of n, a(n) for n=1..54.

FORMULA

a(n) = A001597(n) - A246655(n).

MATHEMATICA

perpowQ[n_]:=n==1||GCD@@FactorInteger[n][[All, 2]]>1;

per=Select[Range[1000], perpowQ];

per-NestList[NestWhile[#+1&, #+1, !PrimePowerQ[#]&]&, 2, Length[per]-1]

PROG

(Python)

from sympy import mobius, primepi, integer_nthroot

def A377044(n):

def bisection(f, kmin=0, kmax=1):

while f(kmax) > kmax: kmax <<= 1

while kmax-kmin > 1:

kmid = kmax+kmin>>1

if f(kmid) <= kmid:

kmax = kmid

else:

kmin = kmid

return kmax

def f(x): return int(n-1+x+sum(mobius(k)*(integer_nthroot(x, k)[0]-1) for k in range(2, x.bit_length())))

def g(x): return int(n+x-sum(primepi(integer_nthroot(x, k)[0]) for k in range(1, x.bit_length())))

return bisection(f, n, n)-bisection(g, n, n) # Chai Wah Wu, Oct 27 2024

CROSSREFS

Including 1 with the prime-powers gives A377043.

A000015 gives the least prime-power >= n.

A000040 lists the primes, differences A001223.

A000961 lists the powers of primes, differences A057820, A093555, A376596.

A001597 lists the perfect-powers, differences A053289, seconds A376559.

A007916 lists the non-perfect-powers, differences A375706, seconds A376562.

A024619 lists the non-prime-powers, differences A375735, seconds A376599.

A025475 lists numbers that are both a perfect-power and a prime-power.

A031218 gives the greatest prime-power <= n.

A080101 counts prime-powers between primes (exclusive).

A106543 lists numbers that are neither a perfect-power nor a prime-power.

A131605 lists perfect-powers that are not prime-powers.

A246655 lists the prime-powers, complement A361102, A375708.

Prime-power runs: A373675, min A373673, max A373674, length A174965.

Prime-power antiruns: A373576, min A120430, max A006549, length A373671.

Cf. A023055, A045542, A052410, A053707, A069623, A110969, A216765, A376560, A376561, A377051.

Sequence in context: A059815 A202670 A203003 * A339427 A319646 A214826

Adjacent sequences: A377041 A377042 A377043 * A377045 A377046 A377047

KEYWORD

sign

AUTHOR

Gus Wiseman, Oct 25 2024

STATUS

approved